One-Bit ExpanderSketch for One-Bit Compressed Sensing

Is it possible to obliviously construct a set of hyperplanes H such that you can approximate a unit vector x when you are given the side on which the vector lies with respect to every h in H? In the sparse recovery literature, where x is approximately k-sparse, this problem is called one-bit compressed sensing and has received a fair amount of attention the last decade. In this paper we obtain the first scheme that achieves almost optimal measurements and sublinear decoding time for one-bit compressed sensing in the non-uniform case. For a large range of parameters, we improve the state of the art in both the number of measurements and the decoding time

On the Power of Adaptivity in Sparse Recovery

The goal of (stable) sparse recovery is to recover a $k$-sparse approximation $x*$ of a vector $x$ from linear measurements of $x$. Specifically, the goal is to recover $x*$ such that ||x-x*||_p <= C min_{k-sparse x'} ||x-x'||_q for some constant $C$ and norm parameters $p$ and $q$. It is known that, for $p=q=1$ or $p=q=2$, this task can be accomplished using $m=O(k \log (n/k))$ non-adaptive measurements [CRT06] and that this bound is tight [DIPW10,FPRU10,PW11]. In this paper we show that if one is allowed to perform measurements that are adaptive, then the number of measurements can be considerably reduced. Specifically, for $C=1+eps$ and $p=q=2$ we show - A scheme with $m=O((1/eps)k log log (n eps/k))$ measurements that uses $O(log* k \log \log (n eps/k))$ rounds. This is a significant improvement over the best possible non-adaptive bound. - A scheme with $m=O((1/eps) k log (k/eps) + k \log (n/k))$ measurements that uses /two/ rounds. This improves over the best possible non-adaptive bound. To the best of our knowledge, these are the first results of this type. As an independent application, we show how to solve the problem of finding a duplicate in a data stream of $n$ items drawn from ${1, 2, ..., n-1}$ using $O(log n)$ bits of space and $O(log log n)$ passes, improving over the best possible space complexity achievable using a single pass.Comment: 18 pages; appearing at FOCS 201

A compressive sensing algorithm for attitude determination

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 29-30).We propose a framework for compressive sensing of images with local distinguishable objects, such as stars, and apply it to solve a problem in celestial navigation. Specifically, let x [epsilon] RN be an N-pixel image, consisting of a small number of local distinguishable objects plus noise. Our goal is to design an m x N measurement matrix A with m << N, such that we can recover an approximation to x from the measurements Ax. We construct a matrix A and recovery algorithm with the following properties: (i) if there are k objects, the number of measurements m is O((klog N)/(log k)), undercutting the best known bound of O(klog(N/k)) (ii) the matrix A is ultra-sparse, which is important when the signal is weak relative to the noise, and (iii) the recovery algorithm is empirically fast and runs in time sub-linear in N. We also present a comprehensive study of the application of our algorithm to attitude determination, or finding one's orientation in space. Spacecraft typically use cameras to acquire an image of the sky, and then identify stars in the image to compute their orientation. Taking pictures is very expensive for small spacecraft, since camera sensors use a lot of power. Our algorithm optically compresses the image before it reaches the camera's array of pixels, reducing the number of sensors that are required.by Rishi Vijay Gupta.M.Eng

Compressive Sensing with Local Geometric Features

We propose a framework for compressive sensing of images with local distinguishable objects, such as stars, and apply it to solve a problem in celestial navigation. Specifically, let x be an N-pixel real-valued image, consisting of a small number of local distinguishable objects plus noise. Our goal is to design an m-by-N measurement matrix A with m << N, such that we can recover an approximation to x from the measurements Ax. We construct a matrix A and recovery algorithm with the following properties: (i) if there are k objects, the number of measurements m is O((k log N)/(log k)), undercutting the best known bound of O(k log(N/k)) (ii) the matrix A is very sparse, which is important for hardware implementations of compressive sensing algorithms, and (iii) the recovery algorithm is empirically fast and runs in time polynomial in k and log(N). We also present a comprehensive study of the application of our algorithm to attitude determination, or finding one's orientation in space. Spacecraft typically use cameras to acquire an image of the sky, and then identify stars in the image to compute their orientation. Taking pictures is very expensive for small spacecraft, since camera sensors use a lot of power. Our algorithm optically compresses the image before it reaches the camera's array of pixels, reducing the number of sensors that are required

Performance analysis of compressive sensing recovery algorithms for image processing using block processing

The modern digital world comprises of transmitting media files like image, audio, and video which leads to usage of large memory storage, high data transmission rate, and a lot of sensory devices. Compressive sensing (CS) is a sampling theory that compresses the signal at the time of acquiring it. Compressive sensing samples the signal efficiently below the Nyquist rate to minimize storage and recoveries back the signal significantly minimizing the data rate and few sensors. The proposed paper proceeds with three phases. The first phase describes various measurement matrices like Gaussian matrix, circulant matrix, and special random matrices which are the basic foundation of compressive sensing technique that finds its application in various fields like wireless sensors networks (WSN), internet of things (IoT), video processing, biomedical applications, and many. Finally, the paper analyses the performance of the various reconstruction algorithms of compressive sensing like basis pursuit (BP), compressive sampling matching pursuit (CoSaMP), iteratively reweighted least square (IRLS), iterative hard thresholding (IHT), block processing-based basis pursuit (BP-BP) based onmean square error (MSE), and peak signal to noise ratio (PSNR) and then concludes with future works